Evolution and Optimum Seeking

Genetic Algorithms 155
p( ∆x)
1
1e-5
1e-10
1e-15
-6 -4 -2 0 2 4 6 8 10 ∆x
p( ∆x)
1
1e-5
1e-10
1e-15
-6 -4 -2 0 2 4 6 8 10 ∆x
Figure 5.16: Probability distributions for GA mutations / left: normal binary
code right: Gray code
cli s. As Figure 5.16 clearly shows for a one dimensional case with x (0) =5,l = 4, and
pm =0:001, the expectation values for changes x are di erent from zero in both cases,
and the distribution is in no case unimodal.
5.3.3 Recombination and Mutation
Innovation during evolutionary processes occurs in two di erent ways, for so-called higher
organisms at least. Only the most early and primitive species operate asexually. People
have often said that GAs can do their work without mutations, which, according to
the schema theorem, always hamper the adaptation or optimization process, and that,
on the other hand, ESs can do their work without recombination. The latter is not
true if self-adaptation of the individual mutation variances and covariancesistowork
properly (see Schwefel, 1987), whereas the former conjecture has been disproved by Back
(1993, 1994a,b). For a GA the probability of containing the correct bits for the global
solution, dispersed over its random start population, is 1 ; L 2 ; ,whichmay be close
enough to 1 for = 50 as population size and L = 1000 as length of the bit string
(actually it is 0:999999999999) however, it cannot be guaranteed that those bits will not
get lost in the course of generations. Whether this happens or not, largely depends on
the problem structure, the phenomenon being called deception (e.g., Whitley, 1991 Page
and Richardson, 1992).
If one looks for recombination e ects within GAs on the level of phenotypes, one
stumbles over the fact that a recombined o spring of two parents that are close together
in the phenotype space may largely deviate from both parental positions there. This